#72258293

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#10390
commutator	C₂³×D₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂⁴×D₄	C₂³
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

2373 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2)(3,4)(6,10,13,18)(7,14)(12,19)(16,17);(2,4)(5,6,11,18)(7,15,14,19)(8,10,9,13)(12,17,20,16);(2,4)(5,7,11,14)(6,12,18,20)(8,16,9,17)(10,15,13,19)

Polycyclic presentation:

2^10|a^2=f,b^2=de,c^2=de,d^2=j,b^a=bd,c^a=cdefg,c^b=cdej,d^a=dh,d^b=di,d^c=dhi,e^a=ehij,e^b=ei,e^c=ehi,e^d=ej,f^b=fhj,f^c=fij,h^c=hj,i^b=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258316, #72258450, #72258474
maximal normal subgroup of	#72258316, #72258450, #72258474
maximal quotient of	#72310297, #72337721

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.4*F.5);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.5*F.6*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);